Is the following claim correct?
$Claim:$ If a sequence $(a_n)$ of real numbers has a convergent subsequence, then it must be bounded.
I intuitively feel that the answer is no since one could (possibly) construct some oscillating sequence/function that, say, on odd values of $n$ would dance in a bounded section of $\mathbb{R}$ and, say, on even values would diverge to $+\infty$. (Additionally, the Bolzano-Weierstrass theorem statement would have been stronger if the claim was true.)
Can someone possibly provide a counter-example?
Best Answer
Of course it is false. Consider the sequence $(0,1,0,2,0,3,0,4,\ldots)$. It has a sequence that converges to $0$ but the sequence itself is unbounded.